This book examines the time vibration of the displacement of a structure due to the internal forces, with no damping or external forcing. Practically, vibrations decay with time but in theory these vibrations do not actually decay. For vibrations due to purely internal forces, the dynamic systems are referred to as conservative systems. The methods of solution adopted for solving non-linear single-degree-of-freedom problems may be extended to multi-degree-of-freedom problems. There are many studies in literature on the application of these methods of solution to linear problems and yet so few have been applied to the non-linear problems. Dynamic vibration equations are of great importance not only for understanding the dynamic motion of structures, but also for providing knowledge of differential equations to mathematicians. Several attempts have been made to study dynamic vibration equations. This book gives a nicer approach to numerical solutions to second order ordinary differential equations. It also gives good examples for learners easy understanding of the techniques used.